Prediction method for monitoring performance of power plant instruments

ABSTRACT

Disclosed is a prediction method for monitoring performance of power plant instruments. The prediction method extracts a principal component of an instrument signal, obtains an optimized constant of a SVR model through a response surface methodology using data for optimization, and trains a model using training data. Therefore, compared to an existing Kernel regression method, accuracy for calculating a prediction value can be improved.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims priority to and the benefit of Korean PatentApplication No. 10-2009-0035254, filed on Apr. 22, 2009, which is hereinincorporated by reference in its entirety.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates generally to a prediction method formonitoring performance of power plant instruments. More particularly,the present invention relates to a prediction method for monitoringperformance of power plant during a nuclear power plant operationcontinually and consistently.

2. Description of the Related Art

In general, all power generating facilities are equipped with aplurality of instruments and obtains various signals in real-time fromthe plurality of instruments to utilize the obtained various signals inpower plant's surveillance and protection systems. Especially, nuclearpower plant's measurement channels, related to a safety system, employmulti-instrument concept so as to guarantee accuracy and reliability ofmeasurement signals and also examine and correct power plant instrumentsat an interval of a nuclear fuel cycle, e.g., approximately 18 months,as read in guidelines for operating technique. All over the world,nuclear power plants have been developing a method for lengtheningmonitoring and correction periods of unnecessarily-performed instrumentcorrection tasks through a Condition Based Monitoring (CBM) method.

FIG. 1 is a block diagram of a conventional instrument performanceregular monitoring system. This system is called an Auto-Associativemodel. Referring to FIG. 1, the conventional instrument performanceregular monitoring system includes a prediction model, a comparisonmodule and a decision logic. The conventional instrument performanceregular monitoring system can monitor drift and malfunction ofinstruments by inputting measuring values into the prediction model,outputting prediction values of the prediction model with respect to aninput measurement values, inputting differences between measurementvalues and the prediction values into the decision logic through thecomparison module and continuously monitoring the instruments.

As a method for calculating a prediction value of an instrument, alinear regression method is the most widely used. This method selectssignals of other instruments that have a high linear correlation withrespect to a signal of an instrument that will be predicted, and obtainsa regression coefficient to allow the sum of squares for error of aprediction value and a measurement value to be the minimum. This methodcan be expressed as the following Equation 1.

ΣE ²=Σ(Y−Y′)²  <Equation 1>

The linear regression method can predict independent variables withrespect to unknown dependent variables once a regression coefficient isdetermined with already-known dependent and independent variables.However, in an existing linear regression method, if dependent variableshave large linear relationships, multicollinearity may occur such thatlarge errors may occur in independent variables with respect to smallnoise included in dependent variables.

A Kernel regression method is a non-parametric regression method thatstores selected measurement data as a memory vector, obtains a weightvalue of Kernel from Euclidean distance of a training data set in amemory vector with respect to a measurement signal set, and applies theweight value to the memory vector to obtain a prediction value of ameasurement instrument without using a parameter such as a regressioncoefficient, a weight value that optimizes correlation of an input andan output like an existing linear regression method, or a neuralnetwork. The non-parametric regression method such like the Kernelregression method has a strong advantage over a model having a nonlinearstate of an input/output relationship and signal noise.

The existing Kernel regression method has a calculation procedure as thefollowing 5 steps.

First Step: Training data are represented with a matrix of Equation 2.

$\begin{matrix}{X = \begin{bmatrix}X_{1,1} & X_{1,2} & \cdots & X_{1,m} \\X_{2,1} & X_{2,2} & \cdots & X_{2,m} \\\vdots & \vdots & \ddots & \vdots \\X_{n_{trn},1} & X_{n_{trn},2} & \cdots & X_{n_{trn},m}\end{bmatrix}} & {< {{Equation}\mspace{14mu} 2} >}\end{matrix}$

where X is a training data matrix stored in a memory vector, n is thenumber of training data, and m is a number of an instrument.

Second Step: The sum of Euclidian distance of training data for a firstinstrument signal set is obtained through the following Equation 3.

$\begin{matrix}{{{d\left( {x_{1},q_{1}} \right)} = \sqrt{\sum\limits_{j}\left( {{x_{1}}_{j} - {q_{1}}_{j}} \right)^{2}}}{{d\left( {x_{2},q_{1}} \right)} = \sqrt{\sum\limits_{j}\left( {{x_{2}}_{j} - {q_{1}}_{j}} \right)^{2}}}\vdots {{d\left( {x_{trn},q_{1}} \right)} = \sqrt{\sum\limits_{j}\left( {{x_{trn}}_{j} - {q_{1}}_{j}} \right)^{2}}}} & {< {{Equation}\mspace{14mu} 3} >}\end{matrix}$

where x is training data, q is test data (or, Query data), trn is anumber of training data, and j is a number of an instrument.

Third step: A weight value with respect to each of training data setsand given test data sets is obtained through the following Equation 4including a Kernel function.

$\begin{matrix}\begin{matrix}{w_{1} = \sqrt{K\left( {d\left( {x_{1},q_{1}} \right)} \right)}} \\{w_{2} = \sqrt{K\left( {d\left( {x_{2},q_{1}} \right)} \right)}} \\\vdots \\\vdots \\{w_{trn} = \sqrt{K\left( {d\left( {x_{trn},q_{1}} \right)} \right)}}\end{matrix} & {< {{Equation}\mspace{14mu} 4} >}\end{matrix}$

In Equation 4, Gaussian Kernel, used as a weight function, is defined asfollows.

${K(d)} = {^{- {(\frac{d^{2}}{\sigma^{2}})}}.}$

Forth step: A prediction value of test data is obtained by multiplying aweight value by each training data and dividing its result by the sum ofweight values as the following Equation 5.

$\begin{matrix}{{{\overset{\Cap}{y}\left( q_{1} \right)} = {\left( {x_{trn},{j*w}} \right)/{\sum w}}}{{\overset{\Cap}{y}\left( q_{1} \right)} = {\left( {\begin{bmatrix}X_{1,1} & X_{1,2} & \ldots & X_{1,j} \\X_{2,1} & X_{2,2} & \ldots & X_{2,j} \\\vdots & \vdots & \ddots & \vdots \\X_{n_{trn},1} & X_{n_{trn},2} & \ldots & X_{n_{trn},j}\end{bmatrix}*\begin{bmatrix}w_{1} \\w_{2} \\\vdots \\w_{tnr}\end{bmatrix}} \right)/{\sum w}}}} & {< {{Equation}\mspace{14mu} 5} >}\end{matrix}$

Fifth step: Second to fourth steps are repeated in order to obtain aprediction value with respect to entire test data.

The existing Kernel regression method has a strong advantage over anonlinear model and signal noise. However, the existing Kernelregression method has disadvantages such as low accuracy due todispersion increase of an output prediction value compared to a linearregression analysis method. The dispersion increase occurs because theAuto-Associative Kernel Regression (AAKR) method stores selectivemeasurement data as a memory vector, obtains a weight value of Kernelfrom Euclidean distance of a training data set in a memory vector withrespect to a measurement signal set, and applies the weight value to thememory vector to obtain a prediction value of an instrument.

SUMMARY OF THE INVENTION

In accordance with one embodiment, a prediction method for monitoringperformance of power plant instruments includes displaying entire datain a matrix, normalizing the entire data into a data set, trisecting thenormalized data set into three data sets, wherein the three data setscomprising a training data set, a optimization data set, and a test dataset, extracting a principal component of each of the normalized trainingdata set, the optimization data set, and the test data set, calculatingan optimal constant of a Support Vector Regression (SVR) model tooptimize prediction value errors of data for optimization using aresponse surface method, generating the Support Vector Regression (SVR)training model using the optimal constant, obtaining a Kernel functionmatrix using the normalized test data set as an input and predicting anoutput value of the support vector regression model, and de-normalizingthe output value into an original range to obtain a predicted value of avariable.

It is therefore a feature of one embodiment to provide a predictionmethod for monitoring performance of plant instruments using a mainfactor analysis and Support Vector Regression (SVR) to improvecalculation accuracy of a prediction value compared to the existingKernel regression method. That is, this method is provided to improveconventional low prediction accuracy. The method uses plant datanormalization, principal component extraction, optimization ofparameters (Kernel bandwidth σ, loss function constant ε, penalty C) ofa SVR model regression method using a response surface methodology,realization of a SVR model through the above, and a de-normalizationmethod of an output prediction value in order to model a plant system,and then monitors instrument signals.

At least one of the above and other features and advantages may berealized by providing a prediction method for monitoring performance ofplant instruments that extracts principal components of power plantdata, creates various case models about a system using a SVR method,optimizes three parameters of a regression Equation using a responsesurface analysis method, and monitors instrument signals after powerplant systems are modeled using the three parameters. Therefore,compared to a widely-used existing Kernel regression method, calculationaccuracy of a prediction value can be improved.

These and other features of the present invention will be more readilyapparent from the detailed description set for the below taken inconjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a conventional instrument performanceregular monitoring system;

FIG. 2 is a flowchart illustrating a prediction method for monitoringperformance of power plant instruments according to one embodiment ofthe present invention;

FIG. 3 illustrates a Support Vector Regression (SVR) model generatedbased on a prediction method for monitoring performance of power plantinstruments according to one embodiment of the present invention;

FIG. 4 illustrates a conceptual diagram of an Optimal Linear Regression(ORL) by SVR;

FIG. 5 illustrates an experimental point of a central composite designwhen the number of model parameters is three;

FIGS. 6A and 6B illustrate reactor core power data in a nuclear powerplant in order to test accuracy according to one embodiment of thepresent invention;

FIGS. 7A and 7B illustrate pressurizer level data in a nuclear powerplant in order to test accuracy according to one embodiment of thepresent invention;

FIGS. 8A and 8B illustrate a steam generator's steam flow data in anuclear power plant in order to test accuracy according to oneembodiment of the present invention;

FIGS. 9A and 9B illustrate a steam generator's narrow range level datain a nuclear power plant in order to test accuracy according to oneembodiment of the present invention;

FIGS. 10A and 10B illustrate a steam generator's pressure data in anuclear power plant in order to test accuracy according to oneembodiment of the present invention;

FIGS. 11A and 11B illustrate a steam generator's wide range level datain a nuclear power plant in order to test accuracy according to oneembodiment of the present invention;

FIGS. 12A and 12B illustrates a steam generator's main feed flow data ina nuclear power plant in order to test accuracy according to oneembodiment of the present invention;

FIGS. 13A and 13B illustrates turbine output data in a nuclear powerplant in order to test accuracy according to one embodiment of thepresent invention;

FIGS. 14A and 14B illustrate first loop charging flow data in a nuclearpower plant in order to test accuracy according to one embodiment of thepresent invention;

FIGS. 15A and 15B illustrate residual heat removing flow data in anuclear power plant in order to test accuracy according to oneembodiment of the present invention; and

FIGS. 16A and 16B illustrate reactor top coolant temperature data in anuclear power plant in order to test accuracy according to oneembodiment of the present invention.

In the following description, the same or similar elements are labeledwith the same or similar reference numbers.

DETAILED DESCRIPTION

The present invention now will be described more fully hereinafter withreference to the accompanying drawings, in which embodiments of theinvention are shown. This invention may, however, be embodied in manydifferent forms and should not be construed as limited to theembodiments set forth herein. Rather, these embodiments are provided sothat this disclosure will be thorough and complete, and will fullyconvey the scope of the invention to those skilled in the art.

The terminology used herein is for the purpose of describing particularembodiments only and is not intended to be limiting of the invention. Asused herein, the singular forms “a”, “an” and “the” are intended toinclude the plural forms as well, unless the context clearly indicatesotherwise. It will be further understood that the terms “comprises”and/or “comprising,” when used in this specification, specify thepresence of stated features, integers, steps, operations, elements,and/or components, but do not preclude the presence or addition of oneor more other features, integers, steps, operations, elements,components, and/or groups thereof.

Unless otherwise defined, all terms (including technical and scientificterms) used herein have the same meaning as commonly understood by oneof ordinary skill in the art to which this invention belongs. It will befurther understood that terms, such as those defined in commonly useddictionaries, should be interpreted as having a meaning that isconsistent with their meaning in the context of the relevant art andwill not be interpreted in an idealized or overly formal sense unlessexpressly so defined herein.

Preferred embodiments of the present invention will now be described indetail with reference to the accompanying drawings.

FIG. 2 is a flowchart illustrating a prediction method for monitoringperformance of power plant instruments according to one embodiment ofthe present invention.

Referring now to FIG. 2, the prediction method for monitoringperformance of power plant instruments includes displaying entire datain a matrix, normalizing the entire data into a data set, trisecting thenormalized data set into three data sets, wherein the three data setscomprising a training data set, a optimization data set, and a test dataset, extracting a principal component of each of the normalized trainingdata set, optimization data set, and test data set, calculating anoptimal constant of a Support Vector Regression (SVR) model to optimizeprediction value errors of data for optimization using a responsesurface method, generating the Support Vector Regression (SVR) trainingmodel using the optimal constant, obtaining a Kernel function matrixusing the normalized test data set as an input and predicting an outputvalue of the support vector regression model, and de-normalizing theoutput value into an original range to obtain a predicted value of avariable.

In an alternative prediction method for monitoring performance of powerplant instruments includes displaying entire data in a matrix,normalizing the entire data into a data set, extracting a principalcomponent of the normalized data set, calculating an optimal constant ofa Support Vector Regression (SVR) model to optimize prediction valueerrors of data for optimization using a response surface method,generating the Support Vector Regression (SVR) model using the optimalconstant, obtaining a Kernel function matrix using the normalized dataset as an input and predicting an output value of the support vectorregression model, and denormalizing the output value into an originalrange to obtain a predicted value of a variable.

The prediction method for monitoring performance of power plantinstruments including the above configuration according to oneembodiment of the present invention is as follows.

First, displaying a matrix in operation S100 displays entire data in amatrix as shown in Equation 6.

$\begin{matrix}\begin{matrix}{X = \begin{bmatrix}X_{1,1} & X_{1,2} & \ldots & X_{1,m} \\X_{2,1} & X_{2,2} & \ldots & X_{2,m} \\\vdots & \vdots & \ddots & \vdots \\X_{{3n},1} & X_{{3n},2} & \ldots & X_{{3n},m}\end{bmatrix}} \\{= \begin{bmatrix}X_{1} & X_{2} & \ldots & X_{m}\end{bmatrix}} \\{X_{ts} = \begin{bmatrix}X_{{{3i} + 1},1} & X_{{{3i} + 1},2} & \ldots & X_{{{3i} + 1},m}\end{bmatrix}} \\{= \begin{bmatrix}X_{{ts}\; 1} & X_{{ts}\; 2} & \ldots & X_{tsm}\end{bmatrix}} \\{X_{tr} = \begin{bmatrix}X_{{{3i} + 2},1} & X_{{{3i} + 2},2} & \ldots & X_{{{3i} + 2},m}\end{bmatrix}} \\{= \begin{bmatrix}X_{{tr}\; 1} & X_{{tr}\; 2} & \ldots & X_{trm}\end{bmatrix}} \\{X_{op} = \begin{bmatrix}X_{{{3i} + 3},1} & X_{{{3i} + 3},2} & \ldots & X_{{{3i} + 3},m}\end{bmatrix}} \\{= \begin{bmatrix}X_{{op}\; 1} & X_{{op}\; 2} & \ldots & X_{opm}\end{bmatrix}}\end{matrix} & {< {{Equation}\mspace{14mu} 6} >}\end{matrix}$

where X is an entire data set. X_(tr), X_(op), and X_(ts) are a data setfor training, a data set for optimization, and a data set for test,respectively. 3n is the entire number of data and m is a number of aninstrument.

Next, performing normalization in operation S200 normalizes entire datausing the following Equation 7.

$\begin{matrix}{Z_{i} = \frac{X_{i} - {\min \left( X_{i} \right)}}{{\max \left( X_{i} \right)} - {\min \left( X_{i} \right)}}} & {< {{Equation}\mspace{14mu} 7} >}\end{matrix}$

where i=1, 2, . . . , 3n

A data set Z of the entire normalized data can be represented by thefollowing Equation 8.

$\begin{matrix}\begin{matrix}{X = \begin{bmatrix}Z_{1,1} & Z_{1,2} & \ldots & X_{1,m} \\X_{2,1} & X_{2,2} & \ldots & X_{2,m} \\\vdots & \vdots & \ddots & \vdots \\Z_{{3n},1} & Z_{{3n},2} & \ldots & Z_{{3n},m}\end{bmatrix}} \\{= \begin{bmatrix}Z_{1} & Z_{2} & \ldots & Z_{m}\end{bmatrix}}\end{matrix} & {< {{Equation}\mspace{14mu} 8} >}\end{matrix}$

Performing separation in operation S300 separates the data set Z intothree parts such as training, optimization, and test. In this embodimentof the present invention, the trisected data are respectively called asZ_(tr), Z_(op), and Z_(ts) and can have an n×m size like the followingEquation 9.

Z_(ts)=[Z_(3i+1,1) Z_(3i+1,2) . . . Z_(3i+1,m)]

Z_(tr)=[Z_(3i+2,1) Z_(3i+2,2) . . . Z_(3i+2,m)]

Z_(op)=[Z_(3i+3,1) Z_(3i+3,2) . . . Z_(3i+3,m)]  <Equation 9>

where i=0, 1, 2, . . . , n−1.

Extracting a principal component in operation S400 extracts principalcomponents of each of the normalized data sets Z_(tr), Z_(op), andZ_(ts). Dispersion, which is a engenvalue of a covariance matrix, ofprincipal components is arranged according to its size, and principalcomponents P_(tr), P_(op), and P_(ts) with respect to Z_(tr), Z_(op),and Z_(ts) are selected until a cumulative sum reaches greater than99.5%, starting with the principal component of the largest percentagedispersion value.

Regarding a method for obtaining principle components, a PrincipalComponent Analysis (PCA) is a useful method for compressing numerousinput variables into a few variables through linear transformation. Thecompressed variable is called as a principal component. The PCA usescorrelation between variables to radiate data of an original dimensioninto a hyperplane of a low dimension where the sum of their squares ismaximized.

Next, a procedure for extracting a principal component is described. Ifinput variables of an m-dimension are called as x₁, x₂, . . . , x_(m),and new variables generated by a linear combination thereof are calledas θ₁, θ₂, . . . , θ_(m), their relationship can be represented byEquation 10 and Equation 11.

$\begin{matrix}{{\theta_{1} = {{q_{11}z_{1}} + {q_{12}z_{2}} + \ldots + {q_{1m}z_{m}}}}{\theta_{2} = {{q_{21}z_{1}} + {q_{22}z_{2}} + \ldots + {q_{2m}z_{m}}}}\cdots {\theta_{m} = {{q_{m\; 1}z_{1}} + {q_{m\; 2}z_{2}} + \ldots + {q_{mm}z_{m}}}}} & {< {{Equation}\mspace{14mu} 10} >} \\{{\Theta = {QZ}}\begin{matrix}{\Theta = \begin{bmatrix}\theta_{1} \\\theta_{2} \\\vdots \\\theta_{m}\end{bmatrix}} \\{Q = \begin{bmatrix}q_{1,1} & q_{2,1} & \ldots & q_{m,1} \\q_{1,2} & q_{2,2} & \ldots & q_{m,2} \\\vdots & \vdots & \ddots & \vdots \\q_{1,m} & q_{2,m} & \ldots & q_{n,m}\end{bmatrix}} \\{= \begin{bmatrix}q_{1} & q_{2} & \ldots & q_{m}\end{bmatrix}} \\{Z = \begin{bmatrix}z_{1} \\z_{2} \\\vdots \\z_{m}\end{bmatrix}}\end{matrix}} & {< {{Equation}\mspace{14mu} 11} >}\end{matrix}$

Here, Q is linear transformation and Z is the normalized entire dataset. θ₁, θ₂, θ_(m) are called as principal components of a linearsystem. For convenience, the first principal component is designated asthe most important principal component. The most important principalcomponent describes the greatest changes of an input variable, that is,a principal component of the largest dispersion. The lineartransformation Q is used for determination in order to satisfyconditions of the following Equation 12 and Equation 13.

q _(i1) ² +q _(i2) ² + . . . +q _(im) ²=1 for i=1, 2, . . . ,m  <Equation 12>

q _(i1) q _(j1) +q _(i2) q _(j2) + . . . q _(im) q _(jm)=0 fori≠j  <Equation 13>

The condition of Equation 12 maintains a scale after transformation andthe condition of Equation 13 removes correlation between variables aftertransformation.

The principal components can be obtained through the following steps.

Step A. An average value of each variable is subtracted from each of thedata sets Z_(tr), Z_(op), and Z_(ts), and this is called as a matrix A.Here, the training data set Z_(tr) is used as one example fordescription and the matrix A is represented by the following Equation14.

A=Z _(tr) − Z _(tr)  <Equation 14>

Step B. An eigenvalue λ of A^(T)A is obtained and a singular value S ofA is obtained using Equations 15 through 17. The eigenvalues λ exceptfor 0 obtained through Equation 15 are arranged in a descending order,and these are called λ₁, λ₂, . . . , λ_(m), respectively.

$\begin{matrix}{{{{A^{T}A} - {\lambda \; I}}} = 0} & {< {{Equation}\mspace{14mu} 15} >} \\{{s_{1} = \sqrt{\lambda_{1}}},{s_{2} = \sqrt{\lambda_{2}}},\ldots \mspace{14mu},{s_{m} = \sqrt{\lambda_{m}}},\left( {\lambda_{1} \geq \lambda_{2} \geq \ldots \geq \lambda_{m}} \right)} & {< {{Equation}\mspace{14mu} 16} >} \\{S = \begin{bmatrix}S_{1} & 0 & \ldots & 0 \\0 & S_{2} & \ldots & 0 \\\vdots & \vdots & \ddots & 0 \\0 & 0 & \ldots & S_{m}\end{bmatrix}} & {< {{Equation}\mspace{14mu} 17} >}\end{matrix}$

where A^(T) denotes the transpose of a matrix A and I is an identitymatrix.

Step C. An eigenvector of AA^(T), which is an n×n matrix, is obtained,and then a unitary matrix U is obtained. An eigenvalue A is obtainedusing Equation 18, and then is substituted into Equation 19 to obtain aneigenvector e₁, e₂, . . . , e_(m) of n×1 with respect to each eigenvalueλ.

|AA ^(T) −λI|=0  <Equation 18>

(AA ^(T) −λI)X=0  <Equation 19>

Step D. Dispersion of each principal component is obtained usingEquation 20.

$\begin{matrix}{\sigma_{p} = \left( \frac{\begin{bmatrix}S_{1} & S_{2} & \ldots & S_{m}\end{bmatrix}}{\sqrt{n - 1}} \right)^{2}} & {< {{Equation}\mspace{14mu} 20} >}\end{matrix}$

Step E. Percentage is obtained by dividing dispersion of each principalcomponent by the sum of dispersions of entire principal components usingEquation 21 and Equation 22.

$\begin{matrix}{\sigma_{p\_ tot} = {{sum}\left( \frac{\begin{bmatrix}S_{1} & S_{2} & \ldots & S_{m}\end{bmatrix}}{\sqrt{n - 1}} \right)}^{2}} & {< {{Equation}\mspace{14mu} 21} >} \\{{\% \mspace{14mu} \sigma_{p}} = {\left( \frac{\sigma_{p}}{\sigma_{p\_ tot}} \right) \times 100}} & {< {{Equation}\mspace{14mu} 22} >}\end{matrix}$

Step F. The p number of principal components up to preferred percentagedispersion, preferably 99.98%, is selected by performing a cumulativecalculation starting with the largest percentage dispersion % σ_(p).

Step G. A principal component is extracted using Equation 23.

Ptr=[S₁e₁ S₂e₂ . . . S_(p)e_(p)]  <Equation 23>

Step H. Principal components with respect to Z_(op) and Z_(ts) can beextracted using the above same steps.

Table 1 illustrates dispersions of extracted principal components.According to this embodiment of the present invention, 7 principalcomponents are used as shown in Table 1. When the 7 principal componentsare used, 99.9% of entire dispersion can be explained. Therefore,information loss by abandoning the remaining principal components isjust 0.02%.

TABLE 1 No. PC Var Cum Cum % 1 0.70234 0.70234 84.12% 2 0.07859 0.7809393.54% 3   0.0.02905 0.80999 97.02% 4 0.01818 0.82816 99.20% 5 0.003570.83173 99.62% 6 0.00226 0.83400 99.89% 7 0.00071 0.83470 99.98% 80.00011 0.83481 99.99% 9 0.00004 0.83485 100.00% 10 0.00002 0.83486100.00% 11 0.00001 0.83488 100.00%

In the following description, SVR modeling is discussed in detail.Compressing input variables of an m-dimension into principal componentsθ₁, θ₂, . . . , θ_(m) of a p-dimension can be represented by thefollowing Equation 24.

$\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{\theta_{1} = {{q_{11}x_{1}} + {q_{12}x_{2}} + \cdots + {q_{1m}x_{m}}}} \\{\theta_{2} = {{q_{21}x_{1}} + {q_{22}x_{2}} + \cdots + {q_{mm}x_{m}}}}\end{matrix} \\\cdots\end{matrix} \\{\theta_{p} = {{q_{p\; 1}x_{1}} + {q_{p\; 2}x_{2}} + \cdots + {q_{pm}x_{m}}}}\end{matrix} & {< {{Equation}\mspace{14mu} 24} >}\end{matrix}$

where p is an integer, which is equal to or less than m.

An Optimum Regression Line (ORL) obtained through SVR with respect tothe k^(th) output can be represented by the following Equation 25.

f _(k)(θ)=w _(K) ^(T) θ+b _(k)  <Equation 25>

where k is 1, 2, . . . , m.

If ε-insensitive Loss Function with respect to a k^(th) output variabley^((k)) is defined by the following Equation 26, an optimizationequation for obtaining an ORL with respect to y^((k)) can be representedby the following Equation 27.

$\begin{matrix}{{L_{k}\left( y_{k} \right)} = \left\{ \begin{matrix}{0,} & {{{{f_{k}(\theta)} - y^{(k)}}} < ɛ_{k}} \\{{{{{f_{k}(\theta)} - y^{(k)}}} - ɛ_{k}},} & {elsewhere}\end{matrix} \right.} & {< {{Equation}\mspace{14mu} 26} >} \\{{{{Minimize}\; {\Phi \left( {w_{k},\xi_{k}} \right)}} = {{\frac{1}{2}w_{k}^{T}w_{k}} + {C_{k}{\sum\limits_{i = 1}^{n}\left( {\xi_{k,i} + \xi_{k,i}^{*}} \right)}}}}{{{s.t.\mspace{14mu} y_{i}^{(k)}} - {w_{k}^{T}\theta_{i}} - b} \leq {ɛ_{k} + \xi_{k,i}}}{{{w_{k}^{T}\theta_{i}} + b - y_{i}^{(k)}} \leq {ɛ_{k} + \xi_{k,i}^{*}}}{ɛ_{k},\xi_{k,i},{{\xi_{k,i}^{*} \geq {0\mspace{14mu} {for}\mspace{14mu} i}} = 1},2,\ldots \mspace{14mu},n}} & {< {{Equation}\mspace{14mu} 27} >}\end{matrix}$

where k is 1, 2, . . . , m in Equations 26 and 27. ξ_(ki) and ξ_(ki)*are slack variables as shown in FIG. 4. FIG. 4 illustrates a conceptualdiagram of an ORL by SVR. Here, θ_(i) is a principal component vectorcorresponding to an i^(th) observation value vector with respect to x,not an i^(th) component of θ.

The optimization problem may be represented as a dual problem by thefollowing Equation 28.

$\begin{matrix}{{{\max \; \lambda_{k}},{\lambda_{k}^{*}\begin{Bmatrix}{{{- \frac{1}{2}}{\sum\limits_{i = 1}^{n}{\sum\limits_{j = 1}^{n}{\left( {\lambda_{k,i} - \lambda_{k,j}^{*}} \right)\theta_{i}^{T}\theta_{j}}}}} +} \\{\sum\limits_{i = 1}^{n}\left\lbrack {{\lambda_{k,i}\left( {y_{i}^{(k)} - ɛ_{k}} \right)} - {\lambda_{k,j}^{*}\left( {y_{i}^{(k)} - ɛ_{k}} \right)}} \right\rbrack}\end{Bmatrix}}}{{{s.t.\mspace{14mu} 0} \leq \lambda_{k,i}},{{\lambda_{k,j}^{*} \leq {C_{k}\mspace{14mu} {for}\mspace{14mu} i}} = 1},2,\ldots \mspace{14mu},n}{{\sum\limits_{i =^{\prime}}^{n}\left( {\lambda_{k,i} - \lambda_{k,j}^{*}} \right)} = 0}} & {< {{Equation}\mspace{14mu} 28} >}\end{matrix}$

where k is 1, 2, . . . , m.

Accordingly, Lagrange multipliers λ_(k,1) and λ_(ki)* are substitutedinto Equation 29 to determine an ORL with respect to a k^(th) outputvariable of Auto Associative Support Vector Regression (AASVR).

$\begin{matrix}{{f_{k}(\theta)} = {{{w_{k}^{*T}\theta} + b_{k}^{*}} = {{\sum\limits_{i = 1}^{n}{\left( {\lambda_{k,i} - \lambda_{k,j}^{*}} \right)\theta_{i}^{T}\theta}} + b_{k}^{*}}}} & {< {{Equation}\mspace{14mu} 29} >}\end{matrix}$

The above mentioned descriptions are procedures for obtaining anoptimized linear regression equation. If the result of nonlinearlymapping from primal data into a space of high-dimension is called avector Φ(•), a function of Equation 30 defined as an inner product ofΦ(•) is called as Kernel.

K(x _(i) ,x _(j))=Φ(x _(i))^(T)Φ(x _(j))  <Equation 30>

When trying to find an ORL in a high dimension space, it is notnecessary to know both of Φ(x_(i)) and Φ(x_(j)) but it is sufficient toknow only the Kernel function K(x_(i), x_(j)). A Gaussian Radial BasisFunction is used in this embodiment. When a Kernel function K(θ_(i),θ)=Φ(θ_(i))^(T) Φ(θ) is used, the optimal nonlinear regression Equationcan be obtained as following Equation 31.

$\begin{matrix}{{f_{k}(\theta)} = {{\sum\limits_{i = 1}^{n}{\left( {\lambda_{k,i} - \lambda_{k,j}^{*}} \right){K\left( {\theta_{i},\theta} \right)}}} + b_{k}^{*}}} & {< {{Equation}\mspace{14mu} 31} >}\end{matrix}$

Here, a bias term can be calculated using arbitrary support vectorsθ_(r) and θ_(s) through the following Equation 32.

$\begin{matrix}{b_{k}^{*} = {{- \frac{1}{2}}{\sum\limits_{i = 1}^{n}{\left( {\lambda_{k,i} - \lambda_{k,j}^{*}} \right)\left\lbrack {{K\left( {\theta_{i},\theta_{r}} \right)} + {K\left( {\theta_{i},\theta_{s}} \right)}} \right\rbrack}}}} & {< {{Equation}\mspace{14mu} 32} >}\end{matrix}$

In relation to the obtained optimal nonlinear regression equation, if aconstant E of a loss function, a penalty C of a dual objective function,and a Radial Basis Function (RBF) are used as a Kernel, it is preferableto provide a Kernel bandwidth a in advance. The total m number of SVRwith respect to each output is obtained by repeating these processes andalso AASVR of FIG. 3 can be constructed accordingly.

Calculating an optimal constant in operation S500 uses a responsesurface method to obtain optimal constants (ε, C, and σ) of the SVRmodel that minimize prediction value errors of optimized data Z_(op).The optimal constants can be obtained through the following steps.

Step A. The SVR model parameters σ, ε, and C are designated as v₁, v₂,and v₃, respectively.

Step B. A search range with respect to each of v₁, v₂, and v₃ isdefined. An appropriate search range can be obtained through a priorexperience or a small scale preliminary experiment. In this embodiment,v₁=0.56486˜1.63514, v₂=0.010534˜0.010534, and v₃=2.1076˜7.9933.

Step C. Upper limits and lower limits of the search range are designatedas U₁, U₂, and U₃, and L₁, L₂, and L₃, respectively, and modelparameters are normalized using the following Equation 33.

$\begin{matrix}{{x_{1} = \frac{v_{1} - \left( {\left( {U_{1} + L_{1}} \right)/2} \right)}{\left( {U_{1} - L_{1}} \right)/2}},{x_{2} = \frac{v_{2} - \left( {\left( {U_{2} + L_{2}} \right)/2} \right)}{\left( {U_{2} - L_{2}} \right)/2}},{x_{3} = \frac{v_{3} - \left( {\left( {U_{3} + L_{3}} \right)/2} \right)}{\left( {U_{3} - L_{3}} \right)/2}}} & {< {{Equation}\mspace{14mu} 33} >}\end{matrix}$

Step D. Experimental points, which are evaluation points of modelperformance, are determined by considering a search range of normalizedmodel parameters x₁, x₂, and x₃. To determine the experimental points, acentral composite design, which is one of statistical experiment plans,can be used. If the experimental points designated by the centralcomposite design are represented with a three-dimensional space, it canbe expressed as shown in FIG. 5.

Step E. The Experimental points by the central composite design includeeight vertexes, one center point, and six axial points. In order toestimate the size of an experimental error, a repetition experiment canbe performed approximately five times. Coordinates of an axial point isdefined as a=2^(3/4)=1.68179.

α=[number of factor experimental points]^(1/4)  <Equation 34>

If the repetition experiment is performed five times at the centerpoint, experimental points of x₁, x₂, and x₃ by the central compositedesign are shown in the following Table 2.

TABLE 2 No X₁ X₂ X₃ 1 −1 −1 −1 2 1 −1 −1 3 −1 1 −1 4 1 1 −1 5 −1 −1 1 61 −1 1 7 −1 1 1 8 1 1 1 9 −1.68179 0 0 10 1.68179 0 0 11 0 −1.68179 0 120 1.68179 0 13 0 0 −1.68179 14 0 0 1.68179 15 0 0 0 16 0 0 0 17 0 0 0 180 0 0 19 0 0 0

Step F. As shown in Table 2, values of the model parameters v₁, v₂, andv₃ are determined to obtain Table 3 (experimental points of v₁, v₂, andv₃ by a central composite design). These are values of model parametersthat will be directly used to obtain a model.

TABLE 3 No V₁ V₂ V₃ 1 0.564855 0.010534 2.1067 2 1.635144 0.0105342.1067 3 0.564856 0.039967 2.1067 4 1.635144 0.039967 2.1067 5 0.5648560.010534 7.9933 6 1.635144 0.010534 7.9933 7 0.564856 0.039967 7.9933 81.635144 0.039967 7.9933 9 0.2 0.02525 5.05 10 2 0.02525 5.05 11 1.10.0005 5.05 12 1.1 0.05 5.05 13 1.1 0.02525 0.1 14 1.1 0.02525 10 15 1.10.02525 5.05 16 1.1 0.02525 5.05 17 1.1 0.02525 5.05 18 1.1 0.02525 5.0519 1.1 0.02525 5.05

Step G. At each experimental point of Table 3, a beta vector and a biasconstant of a SVR model are obtained using Z_(tr) and P_(tr). For this,a data set Z_(tr) is used. Actually, the same model is obtained from No.15 to No. 19 corresponding to the center points.

Step H. A data set P_(op) is inputted to the m number of AASVR in orderto evaluate the accuracy of each model and then a normalized predictionvalue {circumflex over (Z)}_(op) of optimization data is obtained. Fromthis, an accuracy of an output model, which is MSE, is calculatedthrough Equation 35. Since five center points have the same model,P_(op) is divided into five and separate MSE calculation result isobtained from the divided sub data set.

$\begin{matrix}{{MSE} = {\frac{1}{mn}{\sum\limits_{i = 1}^{m}{\sum\limits_{j = 1}^{n}\left( {z_{ij} - {\overset{\Cap}{z}}_{ij}} \right)^{2}}}}} & {< {{Equation}\mspace{14mu} 35} >}\end{matrix}$

Here, z_(ij) is a j^(th) input data of a sensor i among P_(op).{circumflex over (Z)}_(ij) is an estimation value by a model. The modelis obtained using P_(tr). An MSE calculation result by an experiment isshown in the following Table 4.

TABLE 4 No MSE Log(MSE) 1 0.000126 −8.97923 2 0.000100 −9.21034 30.000552 −7.50196 4 0.000538 −7.52765 5 0.000121 −9.01972 6 0.000094−9.27222 7 0.000552 −7.50196 8 0.000541 −7.52209 9 0.001083 −6.82802 100.000263 −8.24336 11 0.000072 −9.53884 12 0.000829 −7.09529 13 0.000542−7.52024 14 0.000265 −8.23578 15 0.000221 −8.41735 16 0.000204 −8.4973917 0.000265 −8.23578 18 0.000294 −8.13193 19 0.000194 −8.54765

Step I. When obtaining response surface, log(MSE) instead of MSE isused. Considering this, response surface between the model parametersx1, x2, and x3 and log(MSE) is estimated. The response surface isassumed with a two-dimensional model like the following Equation 36.

log(MSE)=β₀+β₁x₁+β₂x₂+β₃x₃+β₁₁x₁ ²+β₂₂x₂ ²+β₃₃x₃²+β₁₂x₁x₂+β₁₃x₁x₃+β₂₃x₂x₃+e  <Equation 36>

where e is a random error.

The response surface estimated in this embodiment is as below.

log(MSE)=−8.3492b ₀−0.2131x ₁−0.7716x ₂−0.0952x ₃−0.2010x ₁ ²−0.0753x ₂²+0.0799x ₃ ²

Step J. Optimized conditions of log(MSE), v₁, v₂, and v₃ that minimize eare obtained using an estimated response surface. Since thetwo-dimensional response surface is assumed, the optimized condition isconfirmed through partial derivative. This is, x₁, x₂, and x₃ satisfyingthe following Equation 35 are obtained.

$\begin{matrix}{{\frac{\partial{\log ({MSE})}}{\partial x_{1}} = 0},\mspace{14mu} {\frac{\partial{\log ({MSE})}}{\partial x_{2}} = 0},\mspace{11mu} \; {\frac{\partial{\log ({MSE})}}{\partial x_{3}} = 0}} & {< {{Equation}\mspace{14mu} 37} >}\end{matrix}$

Regarding the response surface obtained through this embodiment, theoptimized condition is (x*₁, x*₂, and x*₃)=(1.5438, −0.6818, 1.5929).

Step K. The optimized condition x*₁, x*₂, and x*₃ is converted into anoriginal unit using the following Equation 38.

$\begin{matrix}\begin{matrix}{{v_{1} = {{\left( \frac{U_{1} - L_{1}}{2} \right)x_{1}} + \left( \frac{U_{1} + L_{1}}{2} \right)}},v_{2}} \\{{= {{\left( \frac{U_{2} - L_{2}}{2} \right)x_{2}} + \left( \frac{U_{2} + L_{2}}{2} \right)}},v_{3}} \\{= {{\left( \frac{U_{3} - L_{3}}{2} \right)x_{3}} + \left( \frac{U_{3} + L_{3}}{2} \right)}}\end{matrix} & {< {{Equation}\mspace{14mu} 38} >}\end{matrix}$

According to this embodiment, v*₁=1.3910=ρ*, v*₂=0.0005=ε*,v*₃=6.7951=C*. Since the predicted log(MSE) under this condition is−9.9446, if this is taken with an exponent and then converted into MSE,it becomes 0.000048.

Generating the SVR training model in operation S600 receives the threeoptimal constants ε*, C*, and σ*, principal components P_(tr) oftraining data obtained from operation S500, and training data of a firstsignal (a first column of Z_(tr)), and then solves an optimizationequation using a quadratic equation. Then, the generating of the SVRtraining model in operation S600 obtains β₁(n×1), which is a differenceof Lagrangian multipliers, and a bias constant b₁ to generate a model ofSVR1 as shown in FIG. 3. Through the same method, the above procedure isrepeated on second to m^(th) sensors to obtain β₂, β₃, . . . , β_(m),and b₂, b₃, . . . , b_(m), such that a model of SVR_(m) is generatedfrom SVR₂ to SVR_(m) so as to construct the SVR model with respect to anentire sensor as shown in FIG. 3.

Predicting an output of a training model in operation S700 uses theprincipal component P_(tr) of training data and the principal componentP_(ts) of test data to obtain a Kernel function matrix K(n×n) ofGaussian Radial Basis Function. Then, the predicting of an output of atraining model in operation S700 uses β₁, which is a difference ofLagrangian multiplier of SVR model obtained from operation S600, and abias constant b₁ to generate an output of SVR₁. Through the same method,the above procedure is repeated on second to m^(th) sensors to obtain amodel prediction value, that is, an output of SVR_(m) from SVR₂.

This is represented by the following Equation 39.

$\begin{matrix}\begin{matrix}{{\hat{Z}}_{B\; 1} = {{K \times \beta_{1}} + b_{1}}} \\{{\hat{Z}}_{B\; 2} = {{K \times \beta_{2}} + b_{2}}} \\\vdots \\{{\hat{Z}}_{B_{m}} = {{K \times \beta_{m}} + b_{m}}}\end{matrix} & {< {{Equation}\mspace{14mu} 39} >}\end{matrix}$

Performing de-normalization in operation S800 denormalizes theprediction value of the normalized experimental data obtained inoperation S700 into an original range so as to obtain a prediction valuefor each sensor of an original scale. This is represented by thefollowing Equation 40.

$\begin{matrix}{{\hat{X}}_{B\; 1} = {{{\hat{Z}}_{B\; 1}\left\{ {{\max \left( X_{B\; 1} \right)} - {\min \left( X_{B\; 1} \right)}} \right\}} + {\min \left( X_{B\; 1} \right)}}} \\{{\hat{X}}_{B\; 2} = {{{\hat{Z}}_{B\; 2}\left\{ {{\max \left( X_{B\; 2} \right)} - {\min \left( X_{B\; 2} \right)}} \right\}} + {\min \left( X_{B\; 2} \right)}}} \\{\mspace{31mu} {\vdots \mspace{250mu} \vdots}} \\{{\hat{X}}_{B\; m} = {{{\hat{Z}}_{B\; m}\left\{ {{\max \left( X_{B\; m} \right)} - {\min \left( X_{B\; m} \right)}} \right\}} + {\min \left( X_{B\; m} \right)}}}\end{matrix}$

In order to confirm excellence of the prediction method for monitoringperformance of plant instruments according to one embodiment of thepresent invention, the prediction method is compared to existing methodsby using instrument signal data, which are measured from the first andsecond loops of nuclear power plant during the startup period. The dataused for analysis are values measured by a total of 11 sensors.

Table 5 illustrates accuracy comparison of a prediction value between aconventional Kernel regression method and a method for monitoring aplant instrument according to one embodiment of the present invention.

TABLE 5 Measure- Mean Square Mean Accuracy ment Error of Square Improve-Signal Measurement Conventional Error of ment Number Objects Method NewMethod Rate (%) 1 Reactor Power 0.6460 0.1020 84.19 2 Pressurizer 0.41600.1155 72.23 Water Level 3 Steam Generator's 0.0004 0.0001 81.21 SteamFlow 4 Steam Generator's 0.1303 0.0683 47.57 Narrow Range Water Level 5Steam Generator's 0.0548 0.0178 67.60 Steam Pressure 6 Steam Generator's0.0269 0.0038 85.83 Wide Range Water Level 7 Steam Generator's 0.00030.0001 58.55 Main Feed Flow 8 Turbine Power 71.0406 15.2597 78.52 9Reactor Coolant 0.0234 0.0025 89.19 Charging Flow 10 Residual Heat0.0389 0.0067 95.17 Removing Flow 11 Reactor Coolant 0.1755 0.0542 69.15Temperature

Data used for analysis of FIG. 5 are the following values measured by atotal of 11 sensors.

1. Reactor power output (%)

2. Pressurizer level (%)

3. Steam generator's steam flow (Mkg/hr)

4. Steam generator's narrow range water level data (%)

5. Steam generator's steam pressure data (kg/cm2)

6. Steam generator's wide range water level data (%)

7. Steam generator's main feed flow data (Mkg/hr) Turbine power data(MWe)

9. Reactor coolant charging flow data (m3/hr)

10. Residual heat removing flow data (m3/hr)

11. Reactor coolant temperature data (° C.)

If the data are used to draw a graph with a time function, the graphsare FIGS. 6A through 16B.

FIGS. 6A and 6B illustrate reactor core power data in a nuclear powerplant. FIG. 6A represents test input data of Equation 6X_(ts1). FIG. 6Brepresents predicted data {circumflex over (X)}_(ts1) using thealgorithm according to embodiments with respect to test input data ofEquation 40.

FIGS. 7A and 7B illustrate pressurizer level data in a nuclear powerplant in order to test accuracy according to one embodiment of thepresent invention. FIG. 7A represents test input data of Equation6X_(ts2). FIG. 7B represents estimated data {circumflex over (X)}_(ts2)predicted using the algorithm according to embodiments with respect totest input data of Equation 40.

FIGS. 8A and 8B illustrate a steam generator's steam flow data in anuclear power plant in order to test accuracy according to oneembodiment of the present invention. FIG. 8A represents test input dataX_(ts3) of Equation 6. FIG. 8B represents estimated data {circumflexover (X)}_(ts3) predicted using the algorithm according to embodimentswith respect to test input data of Equation 40.

FIGS. 9A and 9B illustrate a steam generator's narrow range water leveldata in a nuclear power plant in order to test accuracy according to oneembodiment of the present invention. FIG. 9A represents test input dataX_(ts4) of Equation 6. FIG. 9B represents estimated data {circumflexover (X)}_(ts4) predicted using the algorithm according to embodimentswith respect to test input data of Equation 40.

FIGS. 10A and 10B illustrate a steam generator's steam pressure data ina nuclear power plant in order to test accuracy according to oneembodiment of the present invention. FIG. 10A represents test input dataX_(ts5) of Equation 6. FIG. 10B represents estimated data {circumflexover (X)}_(ts5) predicted using the algorithm according to embodimentswith respect to test input data of Equation 40.

FIGS. 11A and 11B illustrate a steam generator's wide range water leveldata in a nuclear power plant in order to test accuracy according to oneembodiment of the present invention. FIG. 11A represents test input dataX_(ts6) of Equation 6. FIG. 11B represents estimated data {circumflexover (X)}_(ts6) predicted using the algorithm according to embodimentswith respect to test input data of Equation 40.

FIGS. 12A and 12B illustrate a steam generator's main feed flow data ina nuclear power plant in order to test accuracy according to oneembodiment of the present invention. FIG. 12A represents test input dataX_(ts7) of Equation 6. FIG. 12B represents estimated data {circumflexover (X)}_(ts7) predicted using the algorithm according to embodimentswith respect to test input data of Equation 40.

FIGS. 13A and 13B illustrate turbine power data in a nuclear power plantin order to test accuracy according to one embodiment of the presentinvention. FIG. 13A represents test input data X_(tr8) of Equation 6.FIG. 13B represents estimated data {circumflex over (X)}_(ts8) predictedusing the algorithm according to embodiments with respect to test inputdata of Equation 40.

FIGS. 14A and 14B illustrate reactor coolant charging flow data in anuclear power plant in order to test accuracy according to oneembodiment of the present invention. FIG. 14A represents test input dataX_(ts9) of Equation 6. FIG. 14B represents estimated data {circumflexover (X)}_(ts9) predicted using the algorithm according to embodimentswith respect to test input data of Equation 40.

FIGS. 15A and 15B illustrate residual heat removing flow data in anuclear power plant in order to test accuracy according to oneembodiment of the present invention. FIG. 15A represents test input dataX_(ts10) of Equation 6. FIG. 15B represents estimated data {circumflexover (X)}_(ts10) predicted using the algorithm according to embodimentswith respect to test input data of Equation 40.

FIGS. 16A and 16B illustrate reactor coolant temperature data in anuclear power plant in order to test accuracy according to oneembodiment of the present invention. FIG. 16A represents test input dataX_(ts11) of Equation 6. FIG. 16B represents estimated data X predictedusing the algorithm according to embodiments with respect to test inputdata of Equation 40.

In Table 5, accuracy is the most basic barometer when a prediction modelis applied to operational monitoring. The accuracy is commonlyrepresented with a mean square error of a model prediction value and anactual measurement value.

Equation 41 represents accuracy about one instrument.

$\begin{matrix}{A = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left( {{\overset{\Cap}{x}}_{i} - x_{i}} \right)^{2}}}} & {< {{Equation}\mspace{14mu} 41} >}\end{matrix}$

where N is the number of experimental data, {circumflex over (x)}_(i) isan estimation value of the model of i^(th) experimental data, and x_(i)is a measurement value of i^(th) experimental data.

The prediction method for monitoring performance of plant instrumentsaccording to one embodiment of the present invention extracts aprincipal component of an instrument signal, obtains an optimizedconstant of a SVR model through a response surface methodology usingdata for optimization, and trains a model using training data.Therefore, compared to an existing Kernel regression method, accuracyfor calculating a prediction value can be improved.

The drawings and the forgoing description gave examples of the presentinvention. The scope of the present invention, however, is by no meanslimited by these specific examples. Numerous variations, whetherexplicitly given in the specification or not, such as differences instructure, dimension, and use of material, are possible. The scope ofthe invention is at least as broad as given by the following claims.

1. A prediction method for monitoring performance of power plantinstruments comprising: displaying entire data in a matrix; normalizingthe entire data into a data set; trisecting the normalized data set intothree data sets, wherein the three data sets comprising a training dataset, a optimization data set, and a test data set; extracting aprincipal component of each of the normalized training data set, theoptimization data set, and the test data set; calculating an optimalconstant of a Support Vector Regression (SVR) model to optimizeprediction value errors of data for optimization using a responsesurface method; generating the Support Vector Regression (SVR) trainingmodel using the optimal constant; obtaining a Kernel function matrixusing the normalized test data set as an input and predicting an outputvalue of the support vector regression model; and de-normalizing theoutput value into an original range to obtain a predicted value of avariable.
 2. The prediction method of claim 1, wherein the displayingentire data in a matrix is represented by the following equation:$X = {\begin{bmatrix}X_{1,1} & X_{1,2} & \ldots & X_{1,m} \\X_{2,1} & X_{2,2} & \ldots & X_{2,m} \\\vdots & \vdots & \ddots & M \\X_{{3n},1} & X_{{3n},2} & \ldots & X_{{3n},m}\end{bmatrix} = \left\lbrack {{\begin{matrix}X_{1} & X_{2} & \ldots & \left. X_{m} \right\rbrack\end{matrix}X_{ts}} = {\left\lbrack {{X_{{{3i} + 1},1}\mspace{14mu} X_{{{3i} + 1},2}\mspace{14mu} \ldots}\mspace{14mu},X_{{{3i} + 1},m}} \right\rbrack = {{\left\lbrack {X_{{{ts}\; 1}\mspace{14mu}}X_{{ts}\; 2}\mspace{14mu} \ldots \mspace{14mu} X_{tsm}} \right\rbrack X_{tr}} = {\left\lbrack {{X_{{{3i} + 2},1}\mspace{14mu} X_{{{3i} + 2},2}\mspace{14mu} \ldots}\mspace{14mu},X_{{{3i} + 2},m}} \right\rbrack = {{\left\lbrack {X_{{{tr}\; 1}\mspace{11mu}}\mspace{11mu} X_{{tr}\; 2}\mspace{14mu} \ldots \mspace{14mu} X_{trm}} \right\rbrack X_{op}} = {\left\lbrack {{X_{{{3i} + 1},1}\mspace{14mu} X_{{{3i} + 1},2}\mspace{14mu} \ldots}\mspace{14mu},X_{{{3i} + 3},m}} \right\rbrack = \left\lbrack {X_{{{op}\; 1}\mspace{14mu}}X_{{op}\; 2}\mspace{14mu} \ldots \mspace{14mu} X_{opm}} \right\rbrack}}}}}} \right.}$where X is the entire data set and X_(tr), X_(op), and X_(ts) are thedata set for training, the data set for optimization, and the data setfor test, respectively, 3n is the entire number of data, and m is anumber of an instrument.
 3. The prediction method of claim 1, whereinthe normalizing the entire data is performed through the followingEquation:$Z_{i} = \frac{X_{i} - {\min \left( X_{i} \right)}}{{\max \left( X_{i} \right)} - {\min \left( X_{i} \right)}}$where i=1, 2, . . . 3n
 4. The prediction method of claim 1, wherein thetrisecting the normalized data set is performed through the followingequation, having an n×m dimension:Z_(ts)=[Z_(3i+1,1) Z_(3i+1,2) . . . Z_(3i+1,m)]Z_(tr)=[Z_(3i+2,1) Z_(3i+2,2) . . . Z_(3i+2,m)]Z_(op)=[Z_(3i+3,1) Z_(3i+3,2) . . . Z_(3i+3,m)]  <Equation 9> where i=0,1, 2, n−1, the normalized data set being referred to as Z, the data setfor training being referred to as Z_(tr), the data set for optimizationbeing referred to as Z_(op), and the data set for test being referred toas Z_(ts).
 5. The prediction method of claim 1, wherein extracting aprincipal component of each of the normalized training data set, theoptimization data set, and the test data set, dispersion of theprincipal component that is an eigenvalue of a covariance matrix isarranged according to its size, and principal components P_(tr), Z_(op),and P_(ts) with respect to Z_(tr), Z_(op), and Z_(ts) are selected untila cumulative sum reaches greater than 99.5%, starting with the principalcomponent of the largest percentage dispersion value, to extractprincipal components of each of the normalized data sets Z_(tr), Z_(op),and Z_(ts).
 6. The prediction method of claim 5, wherein a matrixobtained by subtracting an average value of each variable from each ofthe data sets Z_(tr), Z_(op), and Z_(ts), is called a matrix A and isrepresented by the following equation:A=Z _(tr)− Z _(tr)
 7. The prediction method of claim 6, wherein aneigenvalue λ of A^(T)A and a singular value S of A are obtained throughthe following equations: A^(T)A − λ I = 0${s_{1} = \sqrt{\lambda_{1}}},{s_{2} = \sqrt{\lambda_{2}}},\ldots \mspace{14mu},{s_{m} = \sqrt{\lambda_{m}}},\left( {\lambda_{1} \geq \lambda_{2} \geq \Lambda \geq \lambda_{m}} \right)$where the eigenvalues λ except for 0 are arranged in a descending orderand these arranged eigenvalues λ are called λ₁, λ₂, . . . , λ_(m),respectively.
 8. The prediction method of claim 7, wherein aneigenvector of AA^(T) that is an n×n matrix is obtained, and then aunitary matrix U is obtained, and an eigenvalue A is obtained throughthe following Equation A and then is substituted into the followingEquation B to obtain an eigenvector e₁, e₂, . . . , e_(m) of n×1 withrespect to each eigenvalue λ:|AA ^(T) −λI|=0  Equation A(AA ^(T) −λI)X=0  Equation B
 9. The prediction method of claim 8,wherein the dispersion of each principal component is obtained throughthe following Equation:$\sigma_{P} - \left( \frac{\left\lbrack {S_{1}\mspace{14mu} S_{2}\mspace{14mu} \Lambda \mspace{14mu} S_{m}} \right\rbrack}{\sqrt{n - 1}} \right)^{2}$10. The prediction method of claim 9, wherein the percentage is obtainedby dividing dispersion of each principal component by the sum ofdispersions of entire principal components through the followingequations:$\sigma_{p\_ tot} = {{sum}\left( \frac{\left\lbrack {S_{1}\mspace{14mu} S_{2}\mspace{14mu} \Lambda \mspace{14mu} S_{m}} \right\rbrack}{\sqrt{n - 1}} \right)}^{2}$${\% \mspace{14mu} \sigma_{p}} = {\left( \frac{\sigma_{p}}{\sigma_{p\_ tot}} \right) \times 100}$11. The prediction method of claim 10, wherein the p number of theprincipal components up to preferred percentage dispersion (for example,99.98%) is selected by performing a cumulative calculation starting withthe largest percentage dispersion % σ_(p).
 12. The prediction method ofclaim 11 wherein the principal component is extracted through thefollowing equation.Ptr=[S₁e₁ S₂e₂ . . . S_(p)e_(p)]
 13. The prediction method of claim 12,wherein compressing input variables x₁, x₂, . . . , x_(m) of anm-dimension into the principal components θ₁, θ₂, . . . , θ_(m) of ap-dimension is performed through the following equations:$\begin{matrix}\begin{matrix}\begin{matrix}{\theta_{1} = {{q_{11}x_{1}} + {q_{12}x_{2}} + \cdots + {q_{1m}x_{m}}}} \\{\theta_{2} = {{q_{21}x_{1}} + {q_{22}x_{2}} + \cdots + {q_{mm}x_{m}}}}\end{matrix} \\\cdots\end{matrix} \\{\theta_{p} = {{q_{p\; 1}x_{1}} + {q_{p\; 2}x_{2}} + \cdots + {q_{pm}x_{m}}}}\end{matrix}$ where p is an integer equal to or less than m.
 14. Theprediction method of claim 13, wherein an Optimum Regression Line (ORL)obtained as the Support Vector Regression (SVR) with respect to a k^(th)output is represented by the following Equation:f _(k)(θ)=w _(K) ^(T) θ+b _(k) where k is 1, 2, . . . , m.
 15. Theprediction method of claim 14, wherein, when an ε-insensitive LossFunction with respect to a k^(th) output variable y^((k)) is defined bythe following equation, and an optimization equation for obtaining anORL with respect to y^((k)) is represented by the following equation:θ₁ = q₁₁x₁ + q₁₂x₂+  …   + q_(1m)x_(m)θ₂ = q₁₂x₁ + q₂₂x₂+  …   + q_(mm)x_(m) …θ_(p) = q_(p 1)x₁ + q_(p 2)x₂+  …   + q_(pm)x_(m) where k is 1,2, . . . , m and ξ_(ki) and ξ_(ki)* are slack variables.
 16. Theprediction method of claim 15, wherein the optimization problem isrepresented as a dual problem by the following equation:${L_{k}\left( y_{k} \right)} = \left\{ {{{\begin{matrix}{0,} & {{{{f_{k}(\theta)} - y^{(k)}}} < ɛ_{k}} \\{{{{f_{k}(\theta)} - y^{(k)}}{- ɛ_{k}}},} & {elsewhere}\end{matrix}{Minimize}\; {\Phi\left( {w_{k},\xi_{k}} \right)}} = {{{\frac{1}{2}w_{k}^{T}w_{k}} + {C_{k}{\sum\limits_{i = 1}^{n}{\left( {\xi_{k,i} + \xi_{k,i}^{*}} \right){s.t.\mspace{14mu} y_{i}^{(k)}}}}} - {w_{k}^{T}\theta_{i}} - b} \leq {ɛ_{k} + {\xi_{k,i}w_{k}^{T}\theta_{i}} + b - y_{i}^{(k)}} \leq {ɛ_{k} + {\xi_{k,i}^{*}ɛ_{k}}}}},\xi_{k,i},{{\xi_{k,i}^{*} \geq {0\mspace{14mu} {for}\mspace{14mu} i}} = 1},2,\ldots \mspace{14mu},n} \right.$where k is 1, 2, . . . , m.
 17. The prediction method of claim 16,wherein Lagrange multipliers λ_(k,1) and λ_(ki)* are substituted intothe following equation to determine an ORL with respect to a k^(th)output variable of Auto Associative Support Vector Regression (AASVR):${f_{k}(\theta)} = {{{w_{k}^{*T}\theta} + b_{k}^{*}} = {{\sum\limits_{i = 1}^{n}{\left( {\lambda_{k,i} - \lambda_{k,j}^{*}} \right)\theta_{i}^{T}\theta}} + b_{k}^{*}}}$18. The prediction method of claim 17, wherein, when the result ofnonlinearly mapping from primal data into a space of high-dimension iscalled a vector Φ(•), an optimal nonlinear regression line of thefollowing equation is obtained using a Kernel function defined as aninner product of Φ(•), that is the result of the nonlinear mapping:${f_{k}(0)} = {{\sum\limits_{i = 1}^{n}{\left( {\lambda_{k,i} - \lambda_{k,j}^{*}} \right){K\left( {\theta_{i},\theta} \right)}}} + b_{k}^{*}}$19. The prediction method of claim 18, wherein a bias term is calculatedusing θ_(r) and θ_(s), which is an arbitrary support vector, through thefollowing equation:$b_{k}^{*} = {{- \frac{1}{2}}{\sum\limits_{i = 1}^{n}{\left( {\lambda_{k,i} - \lambda_{k,j}^{*}} \right)\left\lbrack {{K\left( {\theta_{i},\theta_{r}} \right)} + {K\left( {\theta_{i},\theta_{s}} \right)}} \right\rbrack}}}$20. A prediction method for monitoring performance of power plantinstruments comprising: displaying entire data in a matrix; normalizingthe entire data into a data set; extracting a principal component of thenormalized data set; calculating an optimal constant of a Support VectorRegression (SVR) model to optimize prediction value errors of data foroptimization using a response surface method; generating the SupportVector Regression (SVR) model using the optimal constant; obtaining aKernel function matrix using the normalized data set as an input andpredicting an output value of the support vector regression model; andde-normalizing the output value into an original range to obtain apredicted value of a variable.